TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Bernstein polynomials converge uniformly to every continuous function on $[0, 1]$

Statement

If $f : [0, 1] \to R$ is continuous, then $B_{n} (f) \to f$ uniformly on $[0, 1]$ .

Facts & Assumptions

Given: A continuous function $f : [0, 1] \to R$ and $ε > 0$ .

[L1]

A continuous function on a compact metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[L2]

The Bernstein basis has the zeroth and centred second moment identities (The zeroth, first, and second centred moments of the Bernstein basis).

[L3]

For each positive real $η$ there is a natural $N \geq 1$ with $1 / N < η$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

Proof

technique · direct

1.1

Choose $δ > 0$ such that $∣ f (s) - f (x) ∣ < ε / 2$ whenever $∣ s - x ∣ < δ$ , and choose $M$ with $∣ f ∣ \leq M$ .

L1choose

1.2

On the far part, $(k / n - x)^{2} \geq δ^{2}$ ; hence its total basis weight is at most $x (1 - x) / (n δ^{2}) \leq 1 / (4 n δ^{2})$ by [L2].

L2algebra

2.1

Split the Bernstein sum into $∣ k / n - x ∣ < δ$ and its complement. The near part is at most $ε / 2$ by the zeroth identity.

step 1.1L2algebra

2.2

Choose $n$ so large that $2 M / (4 n δ^{2}) < ε / 2$ . The far part is then below $ε / 2$ , uniformly in $x$ .

step 1.2L3algebra

3.1

The near and far estimates give $∣ B_{n} (f) (x) - f (x) ∣ < ε$ for every $x$ and all sufficiently large $n$ .

step 2.1step 2.2∎

Depends on

Used by

Polynomials are uniformly dense in C([0,1],ℝ) Corollary

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bernstein polynomial (Encyclopedia of Mathematics) (standard reference, not scraped)

Bernstein polynomials converge uniformly to every continuous function on [0,1][0,1]